The existence of solutions that minimize whole classes of objective functions is closely related to the concept of majorization [1]}. Majorization is a formalization of the vague notion that the components of one vector are less spread out (or closer together) than those of another vector. We say that, in this case, the former vector is majorized by the latter. Majorization can be characterized in several ways, one of which is via Schur-convex functions. More precisely, given two vectors \(x,y \in \mathbb {R}^n\) , \(x\)  is majorized by \(y\)  if and only if \(\Phi (x) \le \Phi (y)\)  for all Schur-convex functions \(\Phi \) . Several variants of majorization exist that can also be characterized in terms of better objective values for particular classes of Schur-convex functions. Prominent examples of these are weak submajorization, weak supermajorization, and weak absolute majorization, which are characterized by better objective values for non-decreasing, non-increasing, and element-wise even Schur-convex functions, respectively (see also [1]} and Section REF ).
